ECE111_2010SPRING_HW3_PROFSOLN_[0]

# ECE111_2010SPRING_HW3_PROFSOLN_[0] - The Cooper Union...

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The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Problem Set III: TIme, Frequency & Transform Domains- Part I Solution Key Spring 2010 1. An FIR °lter has impulse response given by: h = f 2 ; ° 6 ; 7 ; 3 g where the underscore denotes n = 0 . (a) Express h ( n ) as a superposition of impulses. h ( n ) = 2 ° ( n ) ° 6 ° ( n ° 1) + 7 ° ( n ° 2) + 3 ° ( n ° 3) (b) Write a di/erence equation with input x , output y . y ( n ) = 2 x ( n ) ° 6 x ( n ° 1) + 7 x ( n ° 2) + 3 x ( n ° 3) (c) Sketch a transversal °lter realization. see attached (d) Write explicit formulas for H ( z ) and H ( ! ) . H ( z ) = 2 ° 6 z ° 1 + 7 z ° 2 + 3 z ° 3 H ( ! ) = 2 ° 6 e ° j! + 7 e ° j 2 ! + 3 e ° j 3 ! 2. Here we design an FIR °lter by zero placement. Let H ( z ) be the transfer function of a 6 th order FIR °lter with H ( z ) = 0 at 0 : 9 e ± j 2 °= 3 , 0 : 8 e ° j°= 4 and a double zero at z = ° 1 . Also, H ( z ) is scaled (multiplied by a constant factor) so that its DC gain is 1 (i.e., H ( z = 1) = 1 ). This is to be applied to an analog signal sampled at 10 MHz .

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